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Use equivalent fractions as a strategy to add and subtract fractions.
We can show fractions in two ways, one as 4’s, i.e., ¼, 2/4, ¾, and another one as 8’s, i.e., 2/8, 4/8, 6/8. Recommended Articles. This has been a guide to Fraction Number in Excel. Here we discuss how to format Fractions in Excel using Home Tab and Format Cell Option along with practical uses of fractions in excel. How to create fractions other than 1/4. How to create fractions in Word. Version note: This article was originally written for Word 97 and 2000 and later updated for Word 2002 and 2003; while I have added some instructions for Word 2007 and above, much of the content is more applicable to Word 2003 and earlier. The Fraction Calculator will reduce a fraction to its simplest form. You can also add, subtract, multiply, and divide fractions, as well as, convert to a decimal and work with mixed numbers and reciprocals. We also offer step by step solutions. Step 2: Click the blue arrow to submit. DA: 84 PA: 91 MOZ Rank: 19. Fraction Calculator - Fraction Calc.
Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
Apply and extend previous understandings of multiplication and division.
Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = (ac)/(bd).
Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
Interpret multiplication as scaling (resizing), by:
Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.
Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1
Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
1Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.
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23
THERE IS ONE RULE for adding or subtracting fractions: The denominators must be the same -- just as in arithmetic.
Add the numerators, and place their sum
over the common denominator.
| Example 1. | 6x + 3 5 | + | 4x − 1 5 | = | 10x + 2 5 |
The denominators are the same. Add the numerators as like terms.
| Example 2. | 6x + 3 5 | − | 4x − 1 5 |
To subtract, change the signs of the subtrahend, and add.
| 6x + 3 5 | − | 4x − 1 5 | = | 6x + 3 − 4x + 1 5 | = | 2x + 4 5 |
Problem 1.
To see the answer, pass your mouse over the colored area.
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Do the problem yourself first!
| a) | x 3 | + | y 3 | = | x + y 3 | b) | 5 x | − | 2 x | = | 3 x |
| c) | x x − 1 | + | x + 1 x − 1 | = | 2x + 1 x − 1 | d) | 3x − 4 x − 4 | + | x − 5 x − 4 | = | 4x − 9 x − 4 |
| e) | 6x + 1 x − 3 | − | 4x + 5 x − 3 | = | 6x + 1 − 4x − 5 x − 3 | = | 2x − 4 x − 3 |
| f) | 2x − 3 x − 2 | − | x − 4 x − 2 | = | 2x − 3 − x + 4 x − 2 | = | x + 1 x − 2 |
Different denominators -- The LCM
To add fractions with different denominators, we must learn how to construct the Lowest Common Multiple of a series of terms.
The Lowest Common Multiple (LCM) of a series of terms
is the smallest product that contains every factor of each term.
For example, consider this series of three terms:
pqprps
We will now construct their LCM -- factor by factor.
To begin, it will have the factors of the first term:
LCM = pq
Moving on to the second term, the LCM must have the factors pr. But it already has the factor p -- therefore, we need add only the factor r:
LCM = pqr
Finally, moving on to the last term, the LCM must contain the factors ps. But again it has the factor p, so we need add only the factor s:
LCM = pqrs.
That product is the Lowest Common Multiple of pq, pr, ps. It is the smallest product that contains each of them as factors.
Example 3. Construct the LCM of these three terms: x, x2, x3.
Solution. The LCM must have the factor x.
LCM = x
But it also must have the factors of x2 -- which are x ·x. Therefore, we must add one more factor of x :
LCM = x2
Finally, the LCM must have the factors of x3, which are x·x·x. Therefore,
LCM = x3.
x3 is the smallest product that contains x, x2, and x3 as factors.
We see that when the terms are powers of a variable -- x, x2, x3 -- then their LCM is the highest power.
Problem 2. Construct the LCM of each series of terms.
| a) | ab, bc, cd. abcd | b) | pqr, qrs, rst. pqrst |
| c) | a, a2, a3, a4. a4 | d) | a2b, ab2. a2b2 |
e) ab, cd. abcd
We will now see what this has to do with adding fractions.
| Example 4. Add: | 3 ab | + | 4 bc | + | 5 cd |
Solution. To add fractions, the denominators must be the same. Therefore, as a common denominator choose the LCM of the original denominators. Choose abcd. Then, convert each fraction to an equivalent fraction with denominator abcd.
It is necessary to write the common denominator only once:
| 3 ab | + | 4 bc | + | 5 cd | = | 3cd + 4ad + 5ab abcd |
To change into an equivalent fraction with denominator abcd, simply multiply ab by the factors it is missing, namely cd. Therefore, we must also multiply 3 by cd. That accounts for the first term in the numerator.
To change into an equivalent fraction with denominator abcd, multiply bc by the factors it is missing, namely ad. Therefore, we must also multiply 4 by ad. That accounts for the second term in the numerator.
To change into an equivalent fraction with denominator abcd, multiply cd by the factors it is missing, namely ab. Therefore, we must also multiply 5 by ab. That accounts for the last term in the numerator.
That is how to add fractions with different denominators.
Each factor of the original denominators must be a factor
of the common denominator.
Problem 3. Add.
| a) | 5 ab | + | 6 ac | = | 5c + 6b abc |
| b) | 2 pq | + | 3 qr | + | 4 rs | = | 2rs + 3ps + 4pq pqrs |
| c) | 7 ab | + | 8 bc | + | 9 abc | = | 7c + 8a + 9 abc |
| d) | 1 a | + | 2 a2 | + | 3 a3 | = | a2 + 2a + 3 a3 |
| e) | 3 a2b | + | 4 ab2 | = | 3b + 4a a2b2 |
| f) | 5 ab | + | 6 cd | = | 5cd + 6ab abcd |
2.3 Changing Form: Fractions And Decimalsmr. Mac's Page Sheet
| g) | _2_ x(x + 2) | + | __3__ (x + 2)(x − 3) | = | 2(x − 3) + 3x x(x + 2)(x − 3) |
| = | _ 2x − 6 + 3x_ x(x + 2)(x − 3) | ||||
| = | _5x − 6_ x(x + 2)(x − 3) |
At the 2nd Level we will see a similar problem, but the denominators will not be factored.
| Problem 4. Add: 1 − | 1 a | + | c + 1 ab | . But write the answer as |
1 − a fraction.
| 1 − | 1 a | + | c + 1 ab | = | 1 − ( | 1 a | − | c + 1 ab | ) |
| = | 1 − | b − (c + 1) ab |
| = | 1 − | b − c − 1 ab |
Example 5. Denominators with no common factors.
| a m | + | b n |
When the denominators have no common factors, their LCM is simply their product, mn.
| a m | + | b n | = | an + bm mn |
The numerator then appears as the result of 'cross-multiplying' :
an + bm
However, that technique will work only when adding two fractions, and the denominators have no common factors.
| Example 6. | 2 x − 1 | − | 1 x |
Solution. These denominators have no common factors -- x is not a factor of x − 1. It is a term. Therefore, the LCM of denominators is their product.
| 2 x − 1 | − | 1 x | = | 2x − (x − 1) (x − 1)x | = | 2x − x + 1 (x − 1)x | = | _x + 1_ (x − 1)x |
Note: The entire x − 1 is being subtracted. Therefore, we write it in parentheses -- and its signs change.
Problem 5.
| a) | x a | + | y b | = | xb + ya ab | b) | x 5 | + | 3x 2 | = | 2x + 15x 10 | = | 17x 10 |
| c) | 6 x − 1 | + | 3 x + 1 | = | 6(x + 1) + 3(x − 1) (x + 1)(x − 1) |
| = | 6x + 6 + 3x − 3 (x + 1)(x − 1) | ||||
| = | _9x + 3_ (x + 1)(x − 1) |
| d) | 6 x − 1 | − | 3 x + 1 | = | 6(x + 1) − 3(x − 1) (x + 1)(x − 1) |
| = | 6x + 6 − 3x + 3 (x + 1)(x − 1) | ||||
| = | _3x + 9_ (x + 1)(x − 1) |
| e) | 3 x − 3 | − | 2 x | = | 3x − 2(x − 3) (x − 3)x |
| = | 3x − 2x + 6 (x − 3)x | ||||
| = | x + 6 (x − 3)x |
| f) | 3 x − 3 | − | 1 x | = | 3x − (x − 3) (x − 3)x |
| = | 3x − x + 3 (x − 3)x | ||||
| = | 2x + 3 (x − 3)x |
| g) | 1 x | + | 2 y | + | 3 z | = | yz + 2xz + 3xy xyz |
| Example 7. Add: a + | b c | . |
Solution. We have to express a with denominator c.
| a | = | ac c | (Lesson 20) |
Therefore,
| a + | b c | = | ac + b c | . |
Problem 6.
| a) | p q | + r | = | p + qr q | b) | 1 x | − 1 | = | 1 − x x |
| c) x − | 1 x | = | x2 − 1 x | d) 1 − | 1 x2 | = | x2 − 1 x2 |
| e) 1 − | 1 x + 1 | = | x + 1 − 1 x + 1 | = | x x + 1 |
| f) 3 + | 2 x + 1 | = | 3x + 3 + 2 x + 1 | = | 3x + 5 x + 1 |
| Problem 7. Write the reciprocal of | 1 2 | + | 1 3 | . |
| [Hint: Only a single fraction | a b | has a reciprocal; it is | b a | .] |
| 1 2 | + | 1 3 | = | 3 + 2 6 | = | 5 6 |
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| Therefore, the reciprocal is | 6 5 | . |
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